Weibull

Weibull refers to Waloddi Weibull, a Swedish engineer and metallurgist (1887–1979) who introduced the Weibull distribution in 1951. He proposed the distribution as a flexible model for material strength and failure times, and the name has since become associated with its use in reliability engineering and life data analysis.

The two-parameter Weibull distribution is defined by a shape parameter k > 0 and a scale parameter λ > 0. Its probability density function is f(x) = (k/λ) (x/λ)^{k-1} exp[-(x/λ)^k] for x ≥ 0, and its cumulative distribution function is F(x) = 1 − exp[-(x/λ)^k] for x ≥ 0. The mean is λ Γ(1 + 1/k) and the variance is λ^2 [Γ(1 + 2/k) − (Γ(1 + 1/k))^2], where Γ denotes the gamma function. The distribution can model increasing, constant, or decreasing failure rates depending on the shape parameter k: k > 1 implies increasing hazard, k = 1 reduces to the exponential distribution with constant hazard, and 0 < k < 1 implies decreasing hazard.

A three-parameter Weibull distribution introduces a location parameter γ, giving F(x) = 1 − exp[−((x − γ)/λ)^k] for x ≥ γ. In practice, parameter estimation is commonly performed by maximum likelihood methods, method of moments, or specialized graphical methods such as Weibull plots, which linearize certain transformations of the data to estimate k and λ.

Applications span reliability engineering, life data analysis, fatigue and materials science, and even modeling of wind speeds and other environmental data, reflecting the distribution’s versatility in representing lifetimes and extreme value behavior.